Integrand size = 8, antiderivative size = 105 \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {4 x}{15 \arccos (a x)^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{15 a \sqrt {\arccos (a x)}}+\frac {8 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a} \] Output:
2/5*(-a^2*x^2+1)^(1/2)/a/arccos(a*x)^(5/2)+4/15*x/arccos(a*x)^(3/2)-8/15*( -a^2*x^2+1)^(1/2)/a/arccos(a*x)^(1/2)+8/15*2^(1/2)*Pi^(1/2)*FresnelC(2^(1/ 2)/Pi^(1/2)*arccos(a*x)^(1/2))/a
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=-\frac {-6 \sqrt {1-a^2 x^2}-2 i e^{i \arccos (a x)} \arccos (a x) (-i+2 \arccos (a x))-4 (-i \arccos (a x))^{3/2} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+e^{-i \arccos (a x)} \arccos (a x) \left (-2+4 i \arccos (a x)-4 e^{i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )}{15 a \arccos (a x)^{5/2}} \] Input:
Integrate[ArcCos[a*x]^(-7/2),x]
Output:
-1/15*(-6*Sqrt[1 - a^2*x^2] - (2*I)*E^(I*ArcCos[a*x])*ArcCos[a*x]*(-I + 2* ArcCos[a*x]) - 4*((-I)*ArcCos[a*x])^(3/2)*ArcCos[a*x]*Gamma[1/2, (-I)*ArcC os[a*x]] + (ArcCos[a*x]*(-2 + (4*I)*ArcCos[a*x] - 4*E^(I*ArcCos[a*x])*(I*A rcCos[a*x])^(3/2)*Gamma[1/2, I*ArcCos[a*x]]))/E^(I*ArcCos[a*x]))/(a*ArcCos [a*x]^(5/2))
Time = 0.60 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5133, 5223, 5133, 5225, 3042, 3785, 3833}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\arccos (a x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 5133 |
| \(\displaystyle \frac {2}{5} a \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}}dx+\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
| \(\Big \downarrow \) 5223 |
| \(\displaystyle \frac {2}{5} a \left (\frac {2 x}{3 a \arccos (a x)^{3/2}}-\frac {2 \int \frac {1}{\arccos (a x)^{3/2}}dx}{3 a}\right )+\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
| \(\Big \downarrow \) 5133 |
| \(\displaystyle \frac {2}{5} a \left (\frac {2 x}{3 a \arccos (a x)^{3/2}}-\frac {2 \left (2 a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}dx+\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}\right )}{3 a}\right )+\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
| \(\Big \downarrow \) 5225 |
| \(\displaystyle \frac {2}{5} a \left (\frac {2 x}{3 a \arccos (a x)^{3/2}}-\frac {2 \left (\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {2 \int \frac {a x}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a}\right )}{3 a}\right )+\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} a \left (\frac {2 x}{3 a \arccos (a x)^{3/2}}-\frac {2 \left (\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {2 \int \frac {\sin \left (\arccos (a x)+\frac {\pi }{2}\right )}{\sqrt {\arccos (a x)}}d\arccos (a x)}{a}\right )}{3 a}\right )+\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle \frac {2}{5} a \left (\frac {2 x}{3 a \arccos (a x)^{3/2}}-\frac {2 \left (\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {4 \int a xd\sqrt {\arccos (a x)}}{a}\right )}{3 a}\right )+\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle \frac {2}{5} a \left (\frac {2 x}{3 a \arccos (a x)^{3/2}}-\frac {2 \left (\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arccos (a x)}}-\frac {2 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{a}\right )}{3 a}\right )+\frac {2 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}\) |
Input:
Int[ArcCos[a*x]^(-7/2),x]
Output:
(2*Sqrt[1 - a^2*x^2])/(5*a*ArcCos[a*x]^(5/2)) + (2*a*((2*x)/(3*a*ArcCos[a* x]^(3/2)) - (2*((2*Sqrt[1 - a^2*x^2])/(a*Sqrt[ArcCos[a*x]]) - (2*Sqrt[2*Pi ]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[a*x]]])/a))/(3*a)))/5
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-Sqrt[1 - c ^2*x^2])*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1 )) Int[x*((a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-(f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] + Simp[f*(m/(b*c*( n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b *ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2 *d + e, 0] && LtQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e , 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (8 \pi \arccos \left (a x \right )^{3} \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-4 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}+2 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a x +3 \sqrt {2}\, \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \sqrt {-a^{2} x^{2}+1}\right )}{15 a \sqrt {\pi }\, \arccos \left (a x \right )^{3}}\) | \(110\) |
Input:
int(1/arccos(a*x)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/15/a*2^(1/2)/Pi^(1/2)*(8*Pi*arccos(a*x)^3*FresnelC(2^(1/2)/Pi^(1/2)*arcc os(a*x)^(1/2))-4*arccos(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*(-a^2*x^2+1)^(1/2)+2*a rccos(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*a*x+3*2^(1/2)*Pi^(1/2)*arccos(a*x)^(1/2) *(-a^2*x^2+1)^(1/2))/arccos(a*x)^3
Exception generated. \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/arccos(a*x)^(7/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\int \frac {1}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \] Input:
integrate(1/acos(a*x)**(7/2),x)
Output:
Integral(acos(a*x)**(-7/2), x)
Exception generated. \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/arccos(a*x)^(7/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/arccos(a*x)^(7/2),x, algorithm="giac")
Output:
integrate(arccos(a*x)^(-7/2), x)
Timed out. \[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \] Input:
int(1/acos(a*x)^(7/2),x)
Output:
int(1/acos(a*x)^(7/2), x)
\[ \int \frac {1}{\arccos (a x)^{7/2}} \, dx=\frac {-\frac {2 \mathit {acos} \left (a x \right )^{3} \left (\int \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}\, x}{\mathit {acos} \left (a x \right )^{3} a^{2} x^{2}-\mathit {acos} \left (a x \right )^{3}}d x \right ) a^{2}}{5}+\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {\mathit {acos} \left (a x \right )}}{5}}{\mathit {acos} \left (a x \right )^{3} a} \] Input:
int(1/acos(a*x)^(7/2),x)
Output:
(2*( - acos(a*x)**3*int((sqrt( - a**2*x**2 + 1)*sqrt(acos(a*x))*x)/(acos(a *x)**3*a**2*x**2 - acos(a*x)**3),x)*a**2 + sqrt( - a**2*x**2 + 1)*sqrt(aco s(a*x))))/(5*acos(a*x)**3*a)